Attractors in Duffing equation The Duffing equation in its full form is
$$\ddot{x} + \delta \dot{x} -\alpha x + \beta x^3 = \gamma \cos(\omega t)$$
Now for specific values of the parameters several attractors exist (or not). Let's assume that $\alpha = \beta = \omega = 1$, $\delta = 0.15$, while $\gamma = 0.2445$. For these values of the parameters the system has two fixed-point attractors and a period-$3$ attractor. The period-$1$ attractors are located at about $(0.815, 0.242)$ and $(−0.933, 0.299)$. The period-$3$ attractor is located at about $(−1.412,−0.137)$, $(−0.354,−0.614)$, and $(0.645,−0.464)$.
My question is the following: How do we obtain the values of the positions of the attractors? Is there a theoretical way or is it done purely numerically? And if so, how?
 A: The best analytical approach is in this case to use averaging techniques for the full nonlinear, weakly damped and weakly forced Duffing equation (i.e. $\beta$ is order 1, but both $\delta$ and $\gamma$ are reasonably small). The technique of averaging is well explained here for small $\beta$, which makes the situation a bit easier, as the 'basis functions' to average over are well-known trigonometric functions.
If $\beta= 1$, those 'basis functions' or 'generating functions' for the unperturbed systems are Jacobi elliptic functions, see DLMF for more information. The averaging technique works exactly the same, only the integrals become a bit more cumbersome. 
In I Kovacic and M J Brennan (eds), The Duffing Equation: Nonlinear Oscillators and their Behaviour, Wiley (2011), ISBN: 978-0-470-71549-9, this is mentioned on page 82. In particular, they give an extensive reference list to find results obtained using this technique and other techniques. To summarise: look at references [15-33] in Kovacic & Brennan, which are listed here.
